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Problem #PRU-108995

Problems Discrete Mathematics Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (area and volume) Systems of points and line segments Systems of points

13–15 3.0

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

Problem #PRU-32135

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–14 3.0

101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (counterexamples)

13–13 2.0

a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?

b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?

Problem #PRU-34962

Problems Methods Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points, lines, and line segments

12–14 2.0

Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?

Problem #PRU-34981

Problems Discrete Mathematics Combinatorics Systems of points and line segments

13–14 3.0

It is known that in a convex $n$ -gon ( $n > 3$ ) no three diagonals pass through one point. Find the number of points (other than the vertex) where pairs of diagonals intersect.

Problem #PRU-34991

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points, lines, and line segments

13–15 4.0

On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.

Problem #PRU-57750

Problems Discrete Mathematics Combinatorics Systems of points and line segments Center of mass Center of mass formula

14–14 2.0

Prove that the medians of the triangle $A B C$ intersect at one point and that point divides the medians in a ratio of $2 : 1$ , counting from the vertex.

Problem #PRU-60381

Problems Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–13 2.0

There are $n$ points on the plane. How many lines are there with endpoints at these points?

Problem #PRU-60382

Problems Discrete Mathematics Combinatorics Systems of points and line segments Systems of points, lines, and line segments

13–15 2.0

On a plane $n$ randomly placed lines are given. What is the number of triangles formed by them?

Problem #PRU-60383

Problems Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (other)

13–14 3.0

On two parallel lines $a$ and $b$ , the points $A_{1}, A_{2}, \dots, A_{m}$ and $B_{1}, B_{2}, \dots, B_{n}$ are chosen, respectively, and all of the segments of the form $A_{i} B_{j}$ , where $1 \leq i \leq m$ , $1 \leq j \leq n$ . How many intersection points will there be if it is known that no three of these segments intersect at one point?